Lecture L13 + L14 – ASTC25Formation of planets

1.Top-down (GGP) hypotheis and its difficulties2. Standard (core-accretion) scenario( a ) From dust to planetesimals - two ways( b ) From planetesimals to planetary cores: how many planetesimals were there?( c ) Safronov number, runaway growth, oligarchic growth of protoplanets

Or some other way??

Only theories involving disks make sense...

Note: the standard scenario on the left also looks likethe r.h.s. pictures….

Major difference:time of formationof giant protoplanets:

3-10 Myr (left panel)0.1 Myr (right panel)

There are two main possible modes of formation of giant gaseousplanets and exoplanets:

# bottom-up, or accumulation scenario for rocky cores (a.k.a. standard theory) predicts formation time ~(3-10) Myr

(V.Safronov, G.Kuiper, A.Cameron)

@ top-down, by accretion disk breakup as a result of gravitational instability of the disk. A.k.a. GGP = Giant Gaseous Protoplanets formation time < 0.1 Myr (I.Kant, G.Kuiper, A.Cameron)

To understand the perceived need for @, we have to considerdisk evolution and observed time scales.

To understand the physics of @, we need to study the stability of disks against self-gravity waves.

Gravitational Instabilityand the Giant GaseousProtoplanet hypothesis

Fragmentation of disks

Self-gravity as a destabilizing force for the epicyclic oscillations (radial excursions) of gas parcels on slightly elliptic orbits

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To study waves in disks, we substitute into the equations of hydrodynamics the wave in a WKBJ (WKB) approximation, also used in quantum mechanics:it assumes that waves are sinusoidal, tightly wrapped,or that kr >>1. All quantities describing the flow of gas in a disk, such as the density and velocity components, are Fourier-analyzed as

Gregor Wentzel (1898-1978) German/American physicist

Hendrick A. Kramers (1894–1952) Dutch physicist 1926

Léon N. Brillouin (1889-1969) French physicist

Harold Jeffreys (1891 –1989) English mathematician, geophysicist, and astronomer, established a general method of approximation of ODEs in 1923

Some history WKB applied to Schrödinger equation (1925)

,

Example of a crest of the spiral wave ~X1 exp[ … ]for k=const >0 , m=2, (omega)=const.

This spiral pattern has constant shape and rotates withan angular speed

equal to (omega)/m

The argument of theexponential function isconstant on the spiralwavecrest

Dispersion Relation for non-axisymmetric waves in disks tight-winding (WKB) local approximation

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Doppler-shifted frequency

epicyclicfrequency

self- gasgravity pressure

In Keplerian disks, i.e.disksaround point-mass objects

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The dispersion relation is, as in all the physics, a relation between the time and spatial frequencies,

Though it looks more frightening than the one describing thesimple harmonic (sinusoidal) sound wave in air:

you can easily convince yourself that in the limit of vanishing surface density in the disk (no self-gravity!) and vanishing epicyclic frequency (no rotation!), the full dispersion relation assumes the above form. So the waves in a non-rotatingmedium w/o gravity are simply pure pressure (sound) waves. The complications due to rotation lead to a spiral shape of a sound wave, or full self-gravitating pressure wave.

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Dispersion Relation in disks with axisymmetric (m=0) waves

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Gravitational stability requirements

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Q

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Local stability of disk, spiral waves may grow

Local linear instability of waves, clumps form,

but their further evolution depends on equation of

state of the gas.

It turns out that even at Q~1.5 there are unstable global modes.

Question: Do we have to worry about self-gravity and instability?

Ans: Yes

Ans: No

z/r~0.1qd>0.1

From: Laughlin & Bodenheimer (2001)

Disk in this SPH simulationinitially had Q~1.8.

The m-armed global spiral modes of the form

grow and compete with each other.

But the waves in a stableQ~2 disk stop growing and do not form small objects (GGPs).

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Clumps forming in a gravitationallyunstable disk(Q < 1)

Recently, Alan Boss revived the half-abandoned idea of disk fragmentation

GGPs?

Two examples of formally unstable disks not willing to form objects immediately Durisen et al. (2003)

Break-up of the disk depends on the equation of state of the gas,and the treatment of boundary conditions.

Simulations of self-gravitating objects forming in the disk (with grid-based hydrodynamics) shows that rapid thermal cooling is crucial

Armitage and Rice (2003)

Disk not allowedto cool rapidly (cooling timescale > 1 P)

Disk allowedto cool rapidly(on dynamical timescale, <0.5 P)

SPH = SmoothedParticle Hydrodynamics

with 1 millionparticles

Mayer, Quinn, Wadsley, Stadel (2003)

Isothermal(infinitely rapid cooling)

GGP (Giant Gaseous Protoplanet) hypothesis= disk fragmentation scenario (A. Cameron in the 1970s)

Main Advantages: forms giant planets quickly, avoids possible timescale paradox; planets tend to form at large distances amenable to imaging.

MAIN DIFFICULTIES:1. Non-axisymmetric and/or non-local spiral modes start developing not only at Q<1 but already when Q decreases to Q~1.5…2 They redistribute mass and heat the disk => increase Q (stabilize disk).

2. Empirically, this self-regulation of the effects of gravity on disk is seenin disk galaxies, all of which have Q~2 and yet don’t split into many baby gallaxies.

3. The only way to force the disk fragmentation is to lower Q~c/Sigma by a factor of 2 in just one orbital period. This seems impossible.

4. Any clumps in disk (a la Boss’ clumps) may in fact shear and disappearrather than form bound objects. Durisen et al. Have found that the equation of state and the correct treatment of boundary conditions are crucial, but couldnot confirm the fragmentation except in the isothermal E.O.S. case.

5. GGP is difficult to apply to Uranus and Neptune; final masses: Brown Dwarfs not GGPs

6. Does not easily explain core masses of planets and exoplanets, nor the chemical correlations (to be discussed in lecture L23)

Standard Accumulation

Scenario

Two-stage accumulation of planets in disks

Mcore=10 ME(?) =>contraction of theatmosphere and inflowof gas from the disk

Planetesimal = solidbody >1 km

(issues not addressedin the standard theoryso far)

Two scenarios proposed for planetesimal formation

Particles settle in a very thinsub-disk, in which Q<1, thengravitational instability in dust layer forms planetesimals

Particles in a turbulent gas not able to achieve Q<1, stick together via non-gravityforces.

gas

Solid particles(dust, meteoroids)

gasQ>1Q<1

How many planetesimals formed in the solar nebula?

How many planetesimals formed in the solar nebula?Gas mass ~ 0.02-0.1 Msun > 2e33g*0.02=4e31 g ~ 1e32 gDust mass ~ 0.5% of that ~ 1e30 g ~ 100 Earth masses(Z=0.02 but some heavy elements don’t condense)

Planetesimal mass, s =1 km = 1e5 cm, ~(4*pi/3) s^3*(1 g/cm^3) => 1e16 g Assuming 100% efficiency of planet formation

N = 1e30g /1e16g = 1e14, s=1 km

N = 1e30g /1e19g = 1e12, s=10 km (at least that many initially)

N = 1e30g /1e22g = 1e8, s=100 km

N = 1e30g /1e25g = 1e5, s=1000 km

N = 1e30g /1e28g = 100, s=10000 km (rock/ice cores & planets)

N = 1e30g /1e29g = 10, (~10 ME rock/ice cores of giant planets)

Gravitational focusing factor

Stopping the runaway growth of planetary cores

Roche lobe radius grows non-linearly

with the mass of the planet, slowing down the growthas the mass (ratio) increases. The Roche lobe radius rL is connected with the size of the Hill-stable disk region via a factor 2*sqrt(3), which, like the size of rL, we derived earlier in lecture L12.

This will allow us to perform a thought experiment and compute the maximum mass to which a planet growsspontaneously by destabilizing further and further regions.

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Isolation mass in different parts of the Minimum Solar Nebula

* Based on Minimum Solar Nebula (Hayashi nebula) = a disk of just enough gas to contain the same amount of condensable dust as the current

planets; total mass ~ 0.02 Msun, mass within 5AU ~0.002 Msun

Conclusions:(1) the inner & outer Solar System are different: critical core=10ME could only be achieved in the outer sol.sys.(2) there was an epoch of giant impacts onto protoplanets when all those semi-isolated ‘oligarchs’ where colliding